advanced process control and global optimization for ... outline control and optimization of complex...

Advanced Process Control and Global Optimization for

Complex Processes

Yu Yang

June 2nd, 2016 UTC Institute for Advanced Systems Engineering

University of Connecticut

2

Research Scope

Regulatory Control (PID)

Planning/Scheduling

StrategicPlanning

Customer

Advanced Process Control (APC)

$

$

$

$

Decision Hierarchy in Process Industry

complexity

[year]

[month]

[week]

[min]

[sec]

3

Outline

Control and Optimization of Complex Chemical Processes

Advanced Process Control for PDE System

Global Optimization under Uncertainty

Model Predictive Control for Nonlinear ODE System

5

Model Predictive Control

Artificial Pancreas

Semiconductor Manufacturing Pharmaceutical Manufacturing

Refinery

6

Nonlinear Model Predictive ControlOptimization

Objective Function

Model Constraints

min t = k+1

k+p

performance

dx/dt =f(x,u) ymin≤ y≤ymax

FuturePast

Predicted performancew/ optimized decision

Real outcome

k k+1 k+ptime

Decision

Real PlantObserver

Decision

Feedforward

NewInformation

Setpoint

Limitations: 1. No robust stability2. Significant online computational demands

• Control

- Deviation from setpoint

T

setpoint setpoint

k p

t k

x t x Q x t x

7

Robust Stability

Robust control Lyapunov function (RCLF: )

, ,

, , , : bounded uncertainties

x t f x f x t g x g x t u t

f x t g x t

Robustly Invariant Set:

TV x x Px

One-step worst-case MPC:

k k+1 k+ptime

FuturePast

Worst-case model

Nominal model

1V x k V x k

RCLF forms the terminal constraint

,

inf sup 0T

u U f g

x P f x f g x g u

8

Robust Control Lyapunov Function

Objective: Enlarging ROA and reducing residual set

ˆ

,

min max

ˆ

min max

ˆ ˆmax

s.t. inf sup 0

0

ˆ

ˆ ˆ max

T

x

TP

x

T

u f g

T T

x

x Px

x Px

x P f x f g x g u

u u u

P

x

x Px x Px

Bilevel Optimization:Fractional programming: Dinkelbach’s methodCoordinate search: Single variable optimization

Yang & Lee, IET control theory and application, 2012Yang & Lee, Journal of Process Control, 2011

Characterize the level of ROA

Characterize the level of residual set

9

-

-

-

-

-

Performance Improvement

Initial Control

Policies

Value Function

Approximation

1( ) min ( , ) ( ( , ))i i

uJ x E r x u J f x u

Off-line policy improvement

On-line Implementation

PID/One-step worst-case MPC

Approximate Dynamic Programming

H. Nosair, Y. Yang & J. M. Lee, Control Engineering Practice, 2010Y. Yang & J. M. Lee, Computers and Chemical Engineering, 2010

arg min ,u

u k J f x u

T

setpoint setpoint( )k p

t k

J x x t x Q x t x

MPC: , 1 , ,u k u k u k p

10

Example

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2386

388

390

392

394

396

398

400

402

CA

TR

ADP

MPC

ROA

Start

MPC ADP

Averagecomputational time

8.6s 0.1s

Average control cost 252.7 230.2

Y. Yang & J. M. Lee, Journal of Process Control, 2012Y. Yang & J. M. Lee, Journal of Process Control, 2013

/

0 0

/

0 0

1 R

R

E RT

A A A A

E RT

R R R A

p p

FC C C k e C

V

QF HT T T k e C

V C C V

Uncertainty

Uncertainty

T

setpoint setpoint

k p

t k

x t x Q x t x

12

PDE Control: Motivation

Heat equation Fluid mechanics Rod pump

Crystallization Lithium battery Engine

13

PDE Control: Application

2 2

2

2 2

, , t ,x z t x z x z tv c

zz t

Sensor: SurfaceEstimation: Downhole load & Displacement

: displacement

: position

: time

x

z

t

Stress difference, Hooke’s law

Acceleration Friction

14

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Dis

pla

cem

ent

(Feet)

Proposed method

Classical mathod

PDE Control: Application

Displacement and load estimation:

0 1 2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Time (s)

Dynam

ic load (

Lbs)

Proposed method

Classical method

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Dis

pla

cem

ent

(Feet)

Classical method

Proposed method

0 1 2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Time (s)

Dynam

ic load (

Lbs)

Classical method

Proposed method

Do

wn

ho

le D

isp

lace

men

t

Do

wn

ho

le L

oad

( Y. Yang & S. Dubljevic, SPE Annual Conference, 2012 )

15

PDE Control: Model Reduction

Spectral method:

My theoretical contributions: Handle the truncation residual in the model predictive control

Controller and observer synthesis by taking residual into account

1

, i i

i

x z t a t z

1

,sn

i i

i

x z t a t z

Truncation

1

, ,sn

i i

i

x z t a t z R z t

State#

FD 2nd order(max error)

FD 4th order(max error)

Fourier spectral(max error)

16 6.13e-1 2.36e-1 2.55e-4

32 1.99e-1 2.67e-2 1.05e-11

64 5.42e-2 1.85e-3 6.22e-13

Spectral Methods in Fluid Dynamics, Canuto, 1988 (Table 1.1 1st order wave equation).

Slow modal state

16

PDE Control: Model Predictive Control

Two-phase flow (linearized K-S equation)

0

2

4

6

8

0

5

10

15

20

-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

x(

,t)

4 2

4 2

1 2

0,

0, 0, , ,

0, 0, , ,

, : boundary inputs,

, : variance of the thin film thickness.

x x xv

t

x t x l t d u t d t

x t x l t u t

u t t

x t

(Y. Yang & S. Dubljevic, Journal of process control, 2013)

Spectral methodGalerkin’s model reductionMPC ( Full state feedback )

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Time (s)

y

0.5 0.6 0.7 0.8 0.9 1-0.56

-0.55

-0.54

-0.53

-0.52

With fast modal output bound

Without fast modal output bound

17

PDE Control: Controller & Observer

Controller and observer synthesis:

1 2 1 2 1 2, , , , , , , , , , u u u u s u s u sn n n n n n n n na a a a a a a a a

1. How to find the order of slow modal states?2. How to design the feedback law and observer gains to obtain a closed-loop

system with large enough region of attraction?

Questions

Slow modal states

ˆ ˆ ˆ1

1

ˆ1

u u u u s u

s s s s

u

u uu

a k a k u k y k y k y k

a k a k u k

u k

L

K Ga k u k

B

B

18


Results (2nd order Parabolic PDE Stabilization)

00.2

0.40.6

0.81

02

46

810

-100

-50

0

50

100

zt

x

(Y. Yang & S. Dubljevic, European journal of control, 2014)

2

2

1

0

, , , ,

0,

1, 0

, 1

0,1

x x xz t b z t c z t dx z t

t zz

x t u t

xt

z

y t x z t z dz

z

ObserverSpectral methodGalerkin’s model reductionOutput feedback controller (LMI)

Boundary sensor

20

Motivation

Metabolic network RefineryWater network

Global Optimization: mixed-integer nonlinear programming (MINLP)

T

,min

s.t. ,

, ,

, 0,1 .

x yc x

Ax b

F x y d

x y

21

Objective: decide the optimal crude oil procurement and refinery operations to maximize the profit, meet market demands and satisfy quality specifications.

Overview

Refinery Optimization under Uncertainty

Market Demand

Profit

Quality

(U.S. Patent Pub. No: US2016/0140448 A1)

22

Flowchart

Model 1

Blending & Splitting (Pooling problem)

Vapor pressure

Octane number

Sensitivity

Sulfur

Viscosity

Multi-mode

J. P. Favennec, Refinery operation and management, Editions TECHNIP, 2001

z x y

23

Uncertainties:

Model 1: Uncertainties

Crude oil yields: www.bp.com

Crude VR yield Sulfur

Skarv 10.3% 0.209%

Saturno 26.7% 0.373%

Plutonio 21.6% 0.2%

Brent 13.6% 0.253%

Hungo 25.5% 0.321%

Polvo 37.2% 0.741%

Zakum 10.9% 0.966%

Basra 26.2% 1.826%

Thunder 20.1% 0.416%

Mars 27.6% 1.216%

Gaussian Distribution: Mean=nominal valueStandard deviation=0.1×nominal value

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Crude1 Sulfur wt%

Pro

babili

ty

=0.0157

Sampling: 120 scenarios with

different VR yield & sulfur content

Scenario 1 VR Sulfur

… …Scenario 2 VR Sulfur




… …

24

Solution: Stochastic Programming

Stochastic programming vs. Deterministic method:

Crude oil

Selection

&

purchase

Crude oil

real quality

(TBP)

Refinery operations:

Cut point temperature,

Flow rate

UncertaintiesStage I Stage II

Deterministic method:

*

Nominalmax Cost Profitd

d d dx

x x x

Stochastic method:

* max Cost Profit Prs

s s i s ix

i

x x x

Pr : Probability of scenario

: Decison of deterministic method

: Decison of stochastic method

i

d

s

i

x

x

1Pr

2Pr

3Pr

4Pr

25

min cost of crude oil purchase product sales operation costs

s.t. Crude oil source restrictions, Material balance,

Unit capacity limits,

Demand requirements,

Quality constraints,

Pooling equations

Two-stage Stochastic Formulation (Model 1):


10 Crude oil

120 Scenarios: different VR yields and sulfur concentration

Discretization of each crude oil quantity:

5000 barrels = 1 lot

We use the non-convex generalized Benders decomposition (NGBD).

Model (120 scenarios):

13061 constraints,

100 binary variables

13800 continuous variables

2760 bilinear terms

( Yang & Barton, AIChE Journal, 2016 )

26

Method: NGBD

Algorithm:

Cutting plane Cutting plane Cutting plane

120

1

min cost of crude oil purchase+ Scenario cost /120

s.t. Crude oil source restrictions

i

i

Crude Crude

1min Scenario

s.t. Scenario constraints

Crude

2min Scenario


120min Scenario


Lower bound solution: MILP

Upper bound solution: NLP Relaxation: LP

27

Results: Profit/Time

Profit/month (18% improvement)

Computational Time

Million $

13.514

14.515

15.516

16.517

17.518

18.5

Stochastic Deterministic

Average profit

GAMS1 C++2 (BARON, Antigone)

Total time 12.2 hours 10 mins No solution in 1 day

1. General Algebraic Modeling System (GAMS)2. Joint work with Rohit Kannan.

28

Model 2: Flowchart

Refinery flowchart

J. P. Favennec, Refinery operation and management, Editions TECHNIP, 2001

29

Decision variables: cut point temperature

Model 2: CDU Model

dis,VR

bot,VR

: Total mole flowrate in the distillate of VR cut

: Total mole flowrate in the bottom (yield) of VR cut.

P

P

,j j jT T T

Crude oil TBP Pseudocomponent

bot, j dis, j

FI equation

, P P

Vapor pressure

Equilibrium ratio

A. M. Alattas, I. E. Grossmann and I. Palou-Rivera, I&EC research, 2011, 2012.

30

Model 2: CDU Model

Fractionation index (FI) equation

dis, ,

,

bot , ,

dis, ,

bot, ,

, ,ini

,

: Mole fraction of pseudocomponent in the distillate of cut

: Mole fraction of pseudocomponent in the bottom of cut

if

FIj i

j i j

j i

j i

j i

r j j i j

s j

xK T

x

x i j

x i j

FI T Tb TFI

FI

,end

, : true boiling point of pseudocomponent if i

j i j

Tb iT Tb T

, s,dis, ,

, ,

bot , ,

,

1 , 0,1 ,

VP, VP : vapor pressure, : equilibrium constant

r j jFI FIj i

j i j i j i j i i

j i

i j

j i j

xK T v K T v v

x

TK T K

P

Gilbert, R.J.H., AIChE Journal, 1966.

31

Model 2: CDU Model

Vapor pressure surrogate model

0.65 0.7 0.75 0.8 0.85 0.9 0.950

0.5

1

1.5

2

2.5

3

3.5

4

Tr

Vapor

Pre

ssure

(atm

)

State equation data

Approximation data

2

, , ,

, 2 , 1 , 0

j i j i j i

j i j i j iVP Tr Tr Tr

,

1.5 3 6

, , , ,

,

1.5 3 6

, , , ,

,

,

VP

5.96 1 1.18 1 0.56 1 1.32 1exp

4.79 1 0.41 1 8.91 1 4.98 1exp

where , : cut,

: pseudocompone

j i

j i j i j i j i

j i

j i j i j i j i

j i

j

j i

ci

Tr

Tr Tr Tr TrPc

Tr

Tr Tr Tr Tr

Tr

TTr j

T

i

nt, : parameterPc

,j j jT T T

32

0 20 40 60 80 100 120300

400

500

600

700

800

900

1000

1100

Vol%

Tem

pra

ture

(K

)

Scenario1 Crude 1

Scenario2 Crude 1

Uncertainty is in the heavy part of true boiling point (TBP) curve

Model 2: Uncertainty

4

4 types of crude oil,

2 TBP curves for each:

2 scenarios

33

Two-stage Stochastic Formulation (Model 2):


4 Crude oil

Discretization of each crude oil

quantity:

5000 barrels = 1 lot

Model (16 scenarios):

10058 constraints,

168 binary variables (128 in CDU model)

8452 continuous variables

8032 bilinear terms

3456 signomial terms:

min cost of crude oil purchase operation costs product sales

s.t. Crude oil source restrictions, Material balance,

Unit capacity limits

,

CDU model,

Demand requirements,

Quality constraints,

Pooling equations

7

jK T 6.9

jK T 4.25

jK T

34

Nominal Scenario

Difficult global optimization problem (ANTIGONE)

0 500 1000 1500 2000 2500 3000 3500 4000 45000

1

2

3

4

5

6

7

8

9

10

Time (s)

Rel

ativ

e ga

p (%

)

Relative gap

Upper bound-Lower bound=Relative gap=

Upper bound

(ANTIGONE: global optimization software, Misner & Floudas)( Yang & Barton, ADCHEM, 2015 )

Preprocessing:Interval reduction

35

Optimization: NGBD

Initialization

LBD>=UBDPB?

Primal bounding problem (LP)

PBP feasible?

Feasibility problem (LP)

Relaxed master problem (MILP)

Dual-PBP-PCR (MILP)

Update: piece, UBDPB

Update LBD

Primal problem (MINLP)

Update: UBD, UBDPB

UBDPB>=UBD

End

New integer realization

Feasibility cut

Optimality cuts: PBP cut, Dual-PBP-PCR cut

Yes

No

Yes

No

Yes

No

\U

UBDPB= min obj Dual-PBP-PCRk

k T

New Features:1. Enhanced interval reduction2. Adaptive piecewise convex relaxation3. Local dual cutting plane4. Capable to handle non-separable case

36

Comparison

Computational time

Time/Iterations Adaptive1 Non-adaptive2

PBP (s) 3932 3896

Dual-PBP-PCR (s) 41572 87009

PP (s) 10686 6094

RMP (s) 148 73

Total time (h) 15.6 26.9

Iteration 126 95

1. Add binary variable if relaxation gap >=10%.2. Use 6 binary variables for piecewise relaxation .

37

Results:

Profit distribution ($/barrel)

4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.50.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Profit $/barrel

Pro

babi

lity

Deterministic

Stochastic

Deterministic

expected profit

Stochastic

expected profit

38

Summary

Research Scope: Advanced process control/Scheduling for complex processes under uncertainties

Methods: MPC, reinforcement learning, spectral method, two-stage stochastic programming (NGBD).

Applications: Rod pump, refining process design,

CSTR control

39

Supplement

Yu Yang

Process Systems Engineering LaboratoryDepartment of Chemical Engineering

Massachusetts Institute of Technology

40


Parabolic PDE (Example: 2nd order)

Eigenfunction analysis 2

2: b c d

zz

A

1

Solve

Let , then

,s

i i

i i

n

i i

i

z z

f z z

x z t a t z

A

2

2

1

0

, , ,

, c c

x z t x z t x z tb c dx

t zz

y t x z t z z dz

0

1,0, , 0

,0

15

x tx t u t

z

x z x t

u t

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Space step

i(z

)

Initial condition

Boundary condition

Input constraintPoint-wise output

41

1

1

1 1

uu

u u

u su s

u

nn

n n

s

n nn n

f

Bdza

Bdz

Bdza u

a

Bdz

aBdz


Decomposition & Boundary transformation:, , ,

: unstable modal state

: stable slow modal state

: stable fast modal state

u s f

u

s

f

a a a a

a

a

a

1, , 0,un 1, , 0

un is boundedfa t

42

PDE Control: MMPC

Modal model predictive control:

(Assume can be measured)

T T T

1 2,

0

min

ft

s su

t

y Qy u W u W

min max ,s f uy y y y y

1 2f f f f fy Ca B u B

1 2s.t. ,s s s s sa a B u B

7, , ,s uu u a a

1 2 ,u u u u ua a B u B

1 2 ,s s s s sy Ca B u B

1 2 ,u u u u uy Ca B u B

,x z t

• Static bound of the output for fast modal states

• Output for unstable modal states

• Control performance

,cy x z t output

,u sa a are known in real time

43


Dynamic upper and lower bounds for: fy k

1 1 1f f fy k y k y k

k

y fy k

fy k

fy k

44


Solve linear matrix inequality (LMI):

T

1

T

2

2

max

T T

1 1 2 2

0 ,

0 0,

,

0, 0.

u u

u

u u u

S K G

S K

K G K u I

S S S S

Convergence:

LMI is guaranteed to be feasible by increasing

the dimension of sa

lim , 0.k

x z k

Feasibility:

, ,u u uK G L

1

0, 0

0 1

0 ,

,

u u u

u u u

u u u u

u u u

K G K

L L

K G K

B B

C

1

0, 0

0 1

0 ,

,

u u u

u u u

u u u u

u u u

K G K

L L

K G K

B B

C

where

45

Model 2: CDU Model

Pseudocomponent method:

Distillate

BottomjT

jT

Bo

ilin

g Te

mp

era

ture

Pseudocomponent

Light key component

Heavy key component

46

Optimization: Range Reduction

Upper bound solution

continuousx

ObjectiveFunction

47

Optimization: Relaxation

Convex/Piecewise convex relaxation

PPobj

c

PBPobj

c

Convex relaxation gap

Piecewise convex relaxation gap

Partition point

48

Optimization: Piecewise Relaxation

How to select the partition variable/point to get tighter cutting plane?

Partition point

Relaxation

If violation ratio

lo lo lo lo

up up up up

lo up lo up

up lo up lo

u xP u x P xP x P

u x P xP x P

u x P xP x P

u x P xP x P

u

threshold,

then one variable in this term is partitioned.

xP

xP

49

Optimization: Local Cutting Plane

Objective: design the dual cutting plane only valid in a specific region .

Benefits:

1. Better bounds for convex relaxation.

2. Extra tight constraints for Dual-PBP-PCR.

,p pc c

, , , , ,, , , p i p i p p p p p i p i p p i pf y c c c c f y c y c

, ,, , , p i p p p p p i p p

i i

f c c c c f c c

91

,

1

Note: 20 2t

p p p t p

t

c d D Q

50

9* 1 *

, , , ,, , , ,

1

, , , , , ,

, , , , , , , , , , , , , , ,

, , ,

min 20 2

s.t. , , , 0, , ,

,

t

s s p p k s p p t px y f b

p k t

s p j s p i s s p k s p p

k

p i j s p j s p i s p j s p i s p j s p i s

p i j

w x f d D Q

G x y f f c c

b f y f y f y

b

, , , , , , , , , , , ,

, , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , ,

, , , , , , , , , ,

,

,

,

,

s p j s p i s p j s p i s p j s p i s



p i j s p j s p i j s p i

i

f y f y f y

b f y f y f y

b f y f y f y

b f b f

, , , ,

,

, , , 0,1 ,

, , .

s

j

n

s p i s p j s s sx y f

i j p


Enhanced PBP-PCR/Dual-PBP-PCR

, , , ,

9* 1 *

, , ,

1

, ,

, , , ,

, , , , , , , , , , , , , , ,

, , ,

min

s.t. 20 2 ,

, ,

, , , 0,

,

sx y f b

t

p k s p p t p

k t

p k s p p

k

s p j s p i s s


p i j

w x

f d D Q

f c c

G x y f

b f y f y f y

b

, , , , , , , , , , , ,

, , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , ,

, , , , , , , , , ,

,

,

,

,

s p j s p i s p j s p i s p j s p i s



p i j s p j s p i j s p i

i

f y f y f y

b f y f y f y

b f y f y f y

b f b f

, , , ,

,

, , , 0,1 ,

, , .

s

j

n

s p i s p j s s sx y f

i j p

51

Old cut (globally valid):

New cut (local valid):


4 4 9

* T * 1

Dual-PBP-PCR , , ,

1 1 1

obj 20 2t

s s p s p p p s p p t

p p t

c Bc BQ d BQ D

* *

, ,

4 4 9* T * 1

E-Dual-PBP-PCR , , ,

1 1 1

4 4

, ,

1 1, 0 , 1

obj 20 2

1

p h p h

t

s s p s p p p s p p t

p p t

s p h s p h

p ih D h D

c Bc BQ d BQ D

M D M D

0, can be determined systematicallysM * *

E-Dual-PBP-PCR Dual-PBP-PCRNote: obj objc c

Variables: ,d D

52

Optimization: Non-separable case

Bilinear term=stage I variable ×stage II variable

130132

134136

138

106108

110112

114

-3.88

-3.86

-3.84

-3.82

-3.8

-3.78

-3.76

-3.74

x 105

Crude4Crude

1

obj L

BP

LBP1LBP2 & LBP2 Cut

LBP1 cut

LBP1: convex relaxationLBP2: Reformation + convex relaxation

advanced process control and global optimization for ... outline control and optimization of complex...

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